402 research outputs found
Beyond Geometry : Towards Fully Realistic Wireless Models
Signal-strength models of wireless communications capture the gradual fading
of signals and the additivity of interference. As such, they are closer to
reality than other models. However, nearly all theoretic work in the SINR model
depends on the assumption of smooth geometric decay, one that is true in free
space but is far off in actual environments. The challenge is to model
realistic environments, including walls, obstacles, reflections and anisotropic
antennas, without making the models algorithmically impractical or analytically
intractable.
We present a simple solution that allows the modeling of arbitrary static
situations by moving from geometry to arbitrary decay spaces. The complexity of
a setting is captured by a metricity parameter Z that indicates how far the
decay space is from satisfying the triangular inequality. All results that hold
in the SINR model in general metrics carry over to decay spaces, with the
resulting time complexity and approximation depending on Z in the same way that
the original results depends on the path loss term alpha. For distributed
algorithms, that to date have appeared to necessarily depend on the planarity,
we indicate how they can be adapted to arbitrary decay spaces.
Finally, we explore the dependence on Z in the approximability of core
problems. In particular, we observe that the capacity maximization problem has
exponential upper and lower bounds in terms of Z in general decay spaces. In
Euclidean metrics and related growth-bounded decay spaces, the performance
depends on the exact metricity definition, with a polynomial upper bound in
terms of Z, but an exponential lower bound in terms of a variant parameter phi.
On the plane, the upper bound result actually yields the first approximation of
a capacity-type SINR problem that is subexponential in alpha
Effective algorithms and protocols for wireless networking: a topological approach
Much research has been done on wireless sensor networks. However, most protocols
and algorithms for such networks are based on the ideal model Unit Disk Graph
(UDG) model or do not assume any model. Furthermore, many results assume the
knowledge of location information of the network. In practice, sensor networks often
deviate from the UDG model significantly. It is not uncommon to observe stable long
links that are more than five times longer than unstable short links in real wireless
networks. A more general network model, the quasi unit-disk graph (quasi-UDG)
model, captures much better the characteristics of wireless networks. However, the
understanding of the properties of general quasi-UDGs has been very limited, which
is impeding the design of key network protocols and algorithms.
In this dissertation we study the properties for general wireless sensor networks
and develop new topological/geometrical techniques for wireless sensor networking.
We assume neither the ideal UDG model nor the location information of the nodes.
Instead we work on the more general quasi-UDG model and focus on figuring out
the relationship between the geometrical properties and the topological properties of
wireless sensor networks. Based on such relationships we develop algorithms that can
compute useful substructures (planar subnetworks, boundaries, etc.). We also present direct applications of the properties and substructures we constructed including routing,
data storage, topology discovery, etc.
We prove that wireless networks based on quasi-UDG model exhibit nice properties
like separabilities, existences of constant stretch backbones, etc. We develop
efficient algorithms that can obtain relatively dense planar subnetworks for wireless
sensor networks. We also present efficient routing protocols and balanced data storage
scheme that supports ranged queries.
We present algorithmic results that can also be applied to other fields (e.g., information
management). Based on divide and conquer and improved color coding
technique, we develop algorithms for path, matching and packing problem that significantly
improve previous best algorithms. We prove that it is unlikely for certain
problems in operation science and information management to have any relatively
effective algorithm or approximation algorithm for them
Hopf algebras and finite tensor categories in conformal field theory
In conformal field theory the understanding of correlation functions can be
divided into two distinct conceptual levels: The analytic properties of the
correlators endow the representation categories of the underlying chiral
symmetry algebras with additional structure, which in suitable cases is the one
of a finite tensor category. The problem of specifying the correlators can then
be encoded in algebraic structure internal to those categories. After reviewing
results for conformal field theories for which these representation categories
are semisimple, we explain what is known about representation categories of
chiral symmetry algebras that are not semisimple. We focus on generalizations
of the Verlinde formula, for which certain finite-dimensional complex Hopf
algebras are used as a tool, and on the structural importance of the presence
of a Hopf algebra internal to finite tensor categories.Comment: 46 pages, several figures. v2: missing text added after (4.5),
references added, and a few minor changes. v3: typos corrected, bibliography
update
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